Griffiths' derivation assumes only that the motion is nonrelativistic (i.e. we can use the simple formula for Larmor's power) and cyclic (AL force is the radiation force averaged over a cycle), so, modulo relativity, it seems rather exact.

Also - is there anything strange in the particle's acceleration increasing exponentially, as one of the AL force equation's solutions permits? Suppose that the area of the EM field is infinite - the particle's energy would increase without limit, but, then, it can constantly draw energy from the EM field, doesn't it? Do we reject the "exponentially increasing acceleration" solution because it violates the energy conservation law or because it is unphysical (i.e. we don't observe things like this)?

And one more thing - is the EM radiation a theoretical necessity needed to avoid violation of the energy conservation law via self-acceleration? If I understand things correctly, a moving particle can accelerate using its own EM field - in time t0 the particle generates a field, and this field affects the particle movement in later time t1 (since the force affecting the particle is calculated in retarded time, so it depends on the motion's parameters a while earlier). Would it lead to infinite increase in particle's energy if there wasn't any radiation?